Critical mass on the Keller-Segel system with signal-dependent motility

We consider the original Keller-Segel system with signal-dependent motility \cite{KS-1971-JTB2}: \begin{equation*} \begin{cases} u_t=\nabla \cdot(\gamma(v)\nabla u-u\phi(v)\nabla v), &x\in \Omega, ~~t>0, v_t=\Delta v+u-v,& x\in \Omega, ~~t>0, u(x,0)=u_0(x),~v(x,0)=v_0(x), & x\in \Omega, \end{cases} \end{equation*} in a bounded domain $\Omega\subset \mathbb{R}^2$ with smooth boundary subject to homogeneous Neumann boundary conditions, where \begin{equation*}\label{KS-1} \phi(v)=(\alpha-1)\gamma'(v). \end{equation*} When $\alpha=0$ and $\gamma(v)=e^{-\chi v}$ with $\chi>0$, by constructing a Lyapunov functional, we find a critical mass $m_*=\frac{4\pi}{\chi}$ such that the global classical solution exists if $M m_*$ and $M\not \in \{\frac{4\pi m}{\chi}: m\in\mathbb{N}^+\}$, where $\mathbb{N}^+$ denotes the set of positive integers and $M=\int_{\Omega} u_0dx$ is the initial cell mass.